To develop a fast, accurate, and precise method to estimate T1 from SPGR signals.

Theory: The magnitude of the noiseless steady state SPGR signal obtained with flip angle $$$\alpha_n$$$ can be described as²: $$x_n= \frac{M_0(1-e^{-TR/T_1})\sin{ {\alpha_n}}}{1-e^{-TR/T_1}\cos{ {\alpha_n}}} [1]$$ with $$$M_0$$$ the equilibrium longitudinal magnetization. While many T1 estimation methods rewrite $$$x_n$$$ in a linearized form to allow for a linear fitting procedure, nonlinear least squares (NLS) estimators find the optimal parameters $$$\hat{M_0},\hat{T_1}$$$ by minimizing $$$J(M_0,T_1) = \sum_{n=1}^N{(x_n(M_0,T_1) - y_n)}^2$$$, where $$$\{{y_n}\}_{n=1,...,N}$$$ is the measured MR signal. Popular but computationally expensive NLS methods to minimize $$$J(\cdot,\cdot)$$$, such as Levenberg-Marquardt (LM) and Gauss-Newton (GN)³, do not typically exploit the particular structure of the SPGR signal. In fact, the SPGR signal model can be rewritten as a quotient of linear functions of new unknown parameters: $$x_n = \frac{\sum_{k=1}^Kc_kp_k(\alpha_n)}{\sum_{k=1}^Kc_kq_k(\alpha_n)} [2] $$ with $$$K=2$$$ and $$\begin{array}\\ c_0=1, \quad p_0(\alpha_n)=0, \quad q_0(\alpha_n)=1,\\c_1=M_0(1-e^{-TR/T_1}), \quad p_1(\alpha_n)=\sin(\alpha_n), \quad q_1(\alpha_n)=0, \\c_2=e^{-TR/T_1}, \quad p_2(\alpha_n)=0,\quad q_2(\alpha_n)=-\cos(\alpha_n) [3]\end{array}$$where the original parameters $$$M_0, T_1$$$ can be related to $$$c_1$$$ and $$$c_2$$$ as $$$T_1 = -TR/\log(c_2)$$$ and $$$M_0 = c_1/(1-c_2)$$$, respectively. Hence, by minimizing $$$J(\cdot,\cdot)$$$ w.r.t. $$$\mathbf{c}$$$, estimates of $$$M_0$$$ and $$$T_1$$$ can be found through simple algebra. The particular structure of the model [2] can now be exploited by applying the method proposed by Dimitrov and Kamenski⁴ . This method, which can be applied to any model where numerator and denominator are both linear functions of the unknown parameters, was shown to be much faster than the LM and GN method and less sensitive to initial values. Equation the gradient of $$$J(\cdot)$$$ with respect of $$$\mathbf{c}$$$ to zero yields, after some algebraic manipulations, $$\mathbf{G(c)c=M(c)} [4] $$ where the parameter-dependent matrix $$$\mathbf{G(c)}$$$ is given by $$\mathbf{G(c)} = {\mathbf{(F_P - R(c)F_Q)}}^T\mathbf{U(c)}(\mathbf{F_P - YF_Q}) [5],$$ with $$\begin{array} \mathbf{p}_k^T = [p_k(\alpha_1), p_k(\alpha_2), ..., p_k(\alpha_n)] ,\\\mathbf{q}_k^T = [q_k(\alpha_1), q_k(\alpha_2), ..., q_k(\alpha_n)],\\\mathbf{F_P}=[\mathbf{p}_1,\mathbf{p}_2], \\\mathbf{F_Q}=[\mathbf{q}_1,\mathbf{q}_2], \\\mathbf{Y}=\mathbf{diag}(y_1,y_2,...,y_N) [6] \end{array}$$ The parameter-dependent matrix $$$\mathbf{U(c)}$$$ and $$$\mathbf{R(c)}$$$ are calculated as $$$\mathbf{U(c)}=\mathbf{diag}(Q_1^2,Q_2^2,...,Q_N^2)$$$ and $$$\mathbf{R(c)}=\mathbf{diag}(R_1,R_2,...,R_N)$$$ with $$\begin{array}\\Q_n= {(1+c_2q_2(\alpha_n))}^{-1} \\R_n= \frac{c_1p_1(\alpha_n)}{1+c_2q_2(\alpha_n)} [7] \end{array}$$

The structure of [4] suggests the following algorithm (NOVIFAST)

1) At $$$t=0$$$, set $$$\mathbf{c}^{(t)}=\mathbf{c}_{ini}$$$

2) Calculate $$$\mathbf{U(\mathbf{c}^{(t)})}$$$, $$$\mathbf{R(\mathbf{c}^{(t)})}$$$ which gives $$$\mathbf{G(\mathbf{c}^{(t)})}$$$ and $$$\mathbf{M(\mathbf{c}^{(t)})}$$$

3) Obtain $$$\mathbf{c}^{(t+1)}={\mathbf{G(\mathbf{c}^{(t)})}}^{-1}\mathbf{M(\mathbf{c}^{(t)})}$$$

4) If $$$\mid J(\mathbf{c}^{(t+1)})-J(\mathbf{c}^{(t)}) \mid < TOL$$$ go to step 6

5) Set $$$t=t+1$$$ and go to step 2

6) $$$\hat{T_1} = -TR/\log(c_2^{(t+1)})$$$

To ensure non-singularity, the inverse of $$$\mathbf{G(\cdot)}$$$ can be replaced by the Moore-Penrose pseudo inverse. Experimentally we have confirmed the convergence results reported by ⁴. We have checked that NOVIFAST converges in only two iterations with Tol = 1e-6. NOVIFAST is therefore considerably faster than LM and GN, which converged in 14 and 8 iterations, respectively. Furthermore, the cost per iteration is low, determined basically by the inversion of $$$\mathbf{G(\cdot)}$$$.

Experiments: Monte Carlo simulations were performed to validate the NOVIFAST estimator in terms of accuracy, precision and computation time. $$$N_{MC} = 10^5$$$ Rician noisy acquisitions were generated with TR/TE= 5/1.1ms, flip angles set⁵ [2,9,19]°, noise standard deviation $$$\sigma = M_0/SNR$$$ and NEX=4 (number of signal averages before magnitude). Two experiments were conducted, one at SNR=80 (Exp.1) and one at SNR=150 (Exp.2). $$$M_0=1$$$ , whereas different T1 values were generated, ranging from 500ms to 2500ms. NOVIFAST was compared against two linear methods: DESPOT-1² and iteratively reweighted linear least squares (IRWLLS)³, as well as two NLS methods: LM and GN. LM, GN and NOVIFAST were initialized equally ($$$M_0=0.8,T_1 = 1000$$$ ms) for the whole range of T1 values. LM and GN were halted using the same stopping criterion as NOVIFAST.

The accuracy (bias) and precision (standard deviation) of the five T1 estimators for Exp.1 are shown in Fig.1 (a) and Fig.1 (b), respectively. The Cramér-Rao Lower Bound⁶ (CRLB) is also shown. Clearly, NOVIFAST outperforms DESPOT-1 and IRWLLS in terms of accuracy and precision. NOVIFAST has a similar relative bias as LM and is always more accurate than GN. While the computational cost of LM and GN is more than 50 times bigger than that of NOVIFAST, NOVIFAST is slightly faster than IRWLLS and speed-wise comparable to DESPOT-1, Fig.1 (c). At high SNR=150 (Exp.2) differences are small, but similar conclusions can be drawn. NOVIFAST outperforms DESPOT-1 in terms of accuracy and precision and has smaller relative bias than IRWLLS. While DESPOT-1, IRWLLS and NOVIFAST require between 0.01 and 0.1 ms to estimate one T1 value, LM and GN require at least 1ms.The proposed NLS T1 estimator NOVIFAST 1) outperforms state-of-the-art linear estimators, such as DESPOT-1 and IRWLLS, in terms of accuracy and precision, while requiring a comparable computation time, 2) is superior to common LM and GN based NLS estimators in terms of computation time.